L11a508

L11a507

L11a509

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a508 at Knotilus!

Link Presentations

[edit Notes on L11a508's Link Presentations]

Planar diagram presentation	X8192 X18,12,19,11 X10,4,11,3 X2,20,3,19 X16,8,17,7 X20,9,21,10 X12,18,7,17 X4,21,5,22 X14,6,15,5 X22,15,13,16 X6,14,1,13
Gauss code	{1, -4, 3, -8, 9, -11}, {5, -1, 6, -3, 2, -7}, {11, -9, 10, -5, 7, -2, 4, -6, 8, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(w-1)\left(u^{2}v^{2}w+u^{2}\left(-v^{2}\right)+u^{2}vw^{2}-2u^{2}vw+u^{2}v-u^{2}w^{2}+u^{2}w-2uv^{2}w+2uv^{2}-2uvw^{2}+3uvw-2uv+2uw^{2}-2uw+v^{2}w-v^{2}+vw^{2}-2vw+v-w^{2}+w\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{7}+4q^{6}-7q^{5}+13q^{4}+q^{-4}-16q^{3}-4q^{-3}+20q^{2}+8q^{-2}-19q-13q^{-1}+18$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-z^{4}a^{-2}+2z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}-2z^{2}+2a^{-2}-2a^{-4}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$z^{7}a^{-7}-3z^{5}a^{-7}+2z^{3}a^{-7}+4z^{8}a^{-6}-15z^{6}a^{-6}+17z^{4}a^{-6}-5z^{2}a^{-6}+a^{-6}z^{-2}-2a^{-6}+5z^{9}a^{-5}-15z^{7}a^{-5}+10z^{5}a^{-5}+2za^{-5}-2a^{-5}z^{-1}+2z^{10}a^{-4}+7z^{8}a^{-4}-41z^{6}a^{-4}+a^{4}z^{4}+47z^{4}a^{-4}-14z^{2}a^{-4}+2a^{-4}z^{-2}-3a^{-4}+12z^{9}a^{-3}-30z^{7}a^{-3}+4a^{3}z^{5}+14z^{5}a^{-3}-2a^{3}z^{3}+2z^{3}a^{-3}+2za^{-3}-2a^{-3}z^{-1}+2z^{10}a^{-2}+14z^{8}a^{-2}+8a^{2}z^{6}-49z^{6}a^{-2}-7a^{2}z^{4}+42z^{4}a^{-2}+2a^{2}z^{2}-11z^{2}a^{-2}+a^{-2}z^{-2}-2a^{-2}+7z^{9}a^{-1}+11az^{7}-3z^{7}a^{-1}-13az^{5}-16z^{5}a^{-1}+4az^{3}+10z^{3}a^{-1}+11z^{8}-15z^{6}+4z^{4}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 15                       1 -1 13                     3   3 11                   4 1   -3 9                 9 3     6 7               8 5       -3 5             12 8         4 3           9 10           1 1         9 10             -1 -1       5 10               5 -3     3 8                 -5 -5   1 5                   4 -7   3                     -3 -9 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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